Affine actions on Lie groups and post-Lie algebra structures

Burde, Dietrich; Dekimpe, Karel; Vercammen, Kim

doi:10.48550/arxiv.1109.0251

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Koszulness Of Cacti and Other Operads1

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2014

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“…The operad PostLie. The operad PostLie was defined and studied in [7,33], and recently appeared in various contexts, see [3,4,5,6]. It is generated by a skew-symmetric operation [•, •] and an operation • • • without any symmetries that satisfy the relations Proof.…”

Section: Koszulness Of Cacti and Other Operadsmentioning

confidence: 99%

Cacti and filtered distributive laws

Dotsenko

Griffin

2014

Algebr. Geom. Topol.

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Motivated by the second author's construction of a classifying space for the group of pure symmetric automorphisms of a free product, we introduce and study a family of topological operads, the operads of based cacti, defined for every pointed topological space $(Y,\bullet)$. These operads also admit linear versions, which are defined for every augmented graded cocommutative coalgebra $C$. We show that the homology of the topological operad of based $Y$-cacti is the linear operad of based $H_*(Y)$-cacti. In addition, we show that for every coalgebra $C$ the operad of based $C$-cacti is Koszul. To prove the latter result, we use the criterion of Koszulness for operads due to the first author, utilising the notion of a filtered distributive law between two quadratic operads. We also present a new proof of that criterion which works over the ground field of arbitrary characteristic.Comment: 30 page

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